Hecke stability and weight 1 modular forms

TL;DR

This paper introduces a Hecke stability method for computing weight 1 modular forms across all characteristics, addressing computational challenges and exploring applications related to the inverse Galois problem.

Contribution

The paper develops a unified Hecke stability approach for weight 1 modular forms, providing conditions to identify genuine modular forms in various characteristics.

Findings

Hecke stability conditions ensure ratios are genuine modular forms.

Method applies uniformly across all characteristics.

Applications to the refined inverse Galois problem.

Abstract

The Galois representations associated to weight $1$ newforms over $\bar{\mathbb{F}}_p$ are remarkable in that they are unramified at $p$, but the computation of weight $1$ modular forms has proven to be difficult. One complication in this setting is that a weight $1$ cusp form over $\bar{\mathbb{F}}_p$ need not arise from reducing a weight $1$ cusp form over $\bar{\mathbb{Q}}$. In this article we propose a unified "Hecke stability method" for computing spaces of weight $1$ modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a finite-dimensional Hecke module of ratios of modular forms must consist of genuine modular forms. We conclude with some applications of the Hecke stability method motivated by the refined inverse Galois problem.